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//
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#ifndef dealii_fe_tools_H
#define dealii_fe_tools_H



#include <deal.II/base/config.h>

#include <deal.II/base/exceptions.h>
#include <deal.II/base/geometry_info.h>
#include <deal.II/base/subscriptor.h>
#include <deal.II/base/symmetric_tensor.h>
#include <deal.II/base/tensor.h>

#include <deal.II/distributed/tria.h>

#include <deal.II/fe/component_mask.h>

#include <deal.II/lac/la_parallel_vector.h>

#include <memory>
#include <string>
#include <vector>


DEAL_II_NAMESPACE_OPEN

// Forward declarations
#ifndef DOXYGEN
template <typename number>
class FullMatrix;
template <int dim>
class Quadrature;
template <int dim, int spacedim>
class FiniteElement;
template <int dim, int spacedim>
DEAL_II_CXX20_REQUIRES((concepts::is_valid_dim_spacedim<dim, spacedim>))
class DoFHandler;
template <int dim>
class FiniteElementData;
template <typename number>
class AffineConstraints;
#endif


/**
 * @addtogroup feall
 * @{
 */


/**
 * This namespace offers interpolations and extrapolations of discrete
 * functions of one @p FiniteElement @p fe1 to another @p FiniteElement @p
 * fe2.
 *
 * It also provides the local interpolation matrices that interpolate on each
 * cell. Furthermore it provides the difference matrix $id-I_h$ that is needed
 * for evaluating $(id-I_h)z$ for e.g. the dual solution $z$.
 *
 * For more information about the <tt>spacedim</tt> template parameter check
 * the documentation of FiniteElement or the one of Triangulation.
 */
namespace FETools
{
  /**
   * A base class for factory objects creating finite elements of a given
   * degree. Derived classes are called whenever one wants to have a
   * transparent way to create a finite element object.
   *
   * This class is used in the FETools::get_fe_by_name() and
   * FETools::add_fe_name() functions.
   */
  template <int dim, int spacedim = dim>
  class FEFactoryBase : public Subscriptor
  {
  public:
    /**
     * Create a FiniteElement and return a pointer to it.
     */
    virtual std::unique_ptr<FiniteElement<dim, spacedim>>
    get(const unsigned int degree) const = 0;

    /**
     * Create a FiniteElement from a quadrature formula (currently only
     * implemented for FE_Q) and return a pointer to it.
     */

    virtual std::unique_ptr<FiniteElement<dim, spacedim>>
    get(const Quadrature<1> &quad) const = 0;

    /**
     * Virtual destructor doing nothing but making the compiler happy.
     */
    virtual ~FEFactoryBase() override = default;
  };

  /**
   * A concrete class for factory objects creating finite elements of a given
   * degree.
   *
   * The class's get() function generates a finite element object of the type
   * given as template argument, and with the degree (however the finite
   * element class wishes to interpret this number) given as argument to
   * get().
   */
  template <class FE>
  class FEFactory : public FEFactoryBase<FE::dimension, FE::space_dimension>
  {
  public:
    /**
     * Create a FiniteElement and return a pointer to it.
     */
    virtual std::unique_ptr<FiniteElement<FE::dimension, FE::space_dimension>>
    get(const unsigned int degree) const override;

    /**
     * Create a FiniteElement from a quadrature formula (currently only
     * implemented for FE_Q) and return a pointer to it.
     */
    virtual std::unique_ptr<FiniteElement<FE::dimension, FE::space_dimension>>
    get(const Quadrature<1> &quad) const override;
  };

  /**
   * @warning In most cases, you will probably want to use
   * compute_base_renumbering().
   *
   * Compute the vector required to renumber the dofs of a cell by component.
   * Furthermore, compute the vector storing the start indices of each
   * component in the local block vector.
   *
   * The second vector is organized such that there is a vector for each base
   * element containing the start index for each component served by this base
   * element.
   *
   * While the first vector is checked to have the correct size, the second
   * one is reinitialized for convenience.
   */
  template <int dim, int spacedim>
  void
  compute_component_wise(const FiniteElement<dim, spacedim> &    fe,
                         std::vector<unsigned int> &             renumbering,
                         std::vector<std::vector<unsigned int>> &start_indices);

  /**
   * Compute the vector required to renumber the dofs of a cell by block.
   * Furthermore, compute the vector storing either the start indices or the
   * size of each local block vector.
   *
   * If the @p bool parameter is true, @p block_data is filled with the start
   * indices of each local block. If it is false, then the block sizes are
   * returned.
   *
   * The vector <tt>renumbering</tt> will be indexed by the standard numbering
   * of local degrees of freedom, namely the first vertex, then the second
   * vertex, after vertices lines, quads, and hexes. For each index, the entry
   * indicates the index which this degree of freedom receives in a numbering
   * scheme, where the first block is numbered completely before the second.
   */
  template <int dim, int spacedim>
  void
  compute_block_renumbering(const FiniteElement<dim, spacedim> &  fe,
                            std::vector<types::global_dof_index> &renumbering,
                            std::vector<types::global_dof_index> &block_data,
                            bool return_start_indices = true);

  /**
   * @name Generation of local matrices
   * @{
   */
  /**
   * Compute the interpolation matrix that interpolates a @p fe1-function to a
   * @p fe2-function on each cell. The interpolation_matrix needs to be of
   * size <tt>(fe2.n_dofs_per_cell(), fe1.n_dofs_per_cell())</tt>.
   *
   * Note, that if the finite element space @p fe1 is a subset of the finite
   * element space @p fe2 then the @p interpolation_matrix is an embedding
   * matrix.
   */
  template <int dim, typename number, int spacedim>
  void
  get_interpolation_matrix(const FiniteElement<dim, spacedim> &fe1,
                           const FiniteElement<dim, spacedim> &fe2,
                           FullMatrix<number> &interpolation_matrix);

  /**
   * Compute the interpolation matrix that interpolates a @p fe1-function to a
   * @p fe2-function, and interpolates this to a second @p fe1-function on
   * each cell. The interpolation_matrix needs to be of size
   * <tt>(fe1.n_dofs_per_cell(), fe1.n_dofs_per_cell())</tt>.
   *
   * Note, that this function only makes sense if the finite element space due
   * to @p fe1 is not a subset of the finite element space due to @p fe2, as
   * if it were a subset then the @p interpolation_matrix would be only the
   * unit matrix.
   */
  template <int dim, typename number, int spacedim>
  void
  get_back_interpolation_matrix(const FiniteElement<dim, spacedim> &fe1,
                                const FiniteElement<dim, spacedim> &fe2,
                                FullMatrix<number> &interpolation_matrix);

  /**
   * Compute the identity matrix minus the back interpolation matrix.
   * The @p difference_matrix will be of size <tt>(fe1.n_dofs_per_cell(),
   * fe1.n_dofs_per_cell())</tt> after this function. Previous content
   * of the argument will be overwritten.
   *
   * This function computes the matrix that transforms a @p fe1 function $z$ to
   * $z-I_hz$ where $I_h$ denotes the interpolation operator from the @p fe1
   * space to the @p fe2 space. This matrix hence is useful to evaluate
   * error-representations where $z$ denotes the dual solution.
   */
  template <int dim, typename number, int spacedim>
  void
  get_interpolation_difference_matrix(const FiniteElement<dim, spacedim> &fe1,
                                      const FiniteElement<dim, spacedim> &fe2,
                                      FullMatrix<number> &difference_matrix);

  /**
   * Compute the local $L^2$-projection matrix from fe1 to fe2.
   */
  template <int dim, typename number, int spacedim>
  void
  get_projection_matrix(const FiniteElement<dim, spacedim> &fe1,
                        const FiniteElement<dim, spacedim> &fe2,
                        FullMatrix<number> &                matrix);

  /**
   * This is a rather specialized function used during the construction of
   * finite element objects. It is used to build the basis of shape functions
   * for an element, given a set of polynomials and interpolation points. The
   * function is only implemented for finite elements with exactly @p dim
   * vector components. In particular, this applies to classes derived from
   * the FE_PolyTensor class.
   *
   * Specifically, the purpose of this function is as follows: FE_PolyTensor
   * receives, from its derived classes, an argument that describes a polynomial
   * space. This space may be parameterized in terms of monomials, or in some
   * other way, but is in general not in the form that we use for finite
   * elements where we typically want to use a basis that is derived from
   * some kind of node functional (e.g., the interpolation at specific points).
   * Concretely, assume that the basis used by the polynomial space is
   * $\{\tilde\varphi_j(\mathbf x)\}_{j=1}^N$, and that the node functionals
   * of the finite element are $\{\Psi_i\}_{i=1}^N$. We then want to compute a
   * basis $\{\varphi_j(\mathbf x)\}_{j=1}^N$ for the finite element space so
   * that $\Psi_i[\varphi_j] = \delta_{ij}$. To do this, we can set
   * $\varphi_j(\mathbf x) = \sum_{k=1}^N c_{jk} \tilde\varphi_k(\mathbf x)$
   * where we need to determine the expansion coefficients $c_{jk}$. We do this
   * by applying $\Psi_i$ to both sides of the equation, to obtain
   * @f{align*}{
   *   \Psi_i [\varphi_j] = \sum_{k=1}^N c_{jk} \Psi_i[\tilde\varphi_k],
   * @f}
   * and we know that the left hand side equals $\delta_{ij}$.
   * If you think of this as a system of $N\times N$ equations for the
   * elements of a matrix on the left and on the right, then this can be
   * written as
   * @f{align*}{
   *   I = C X^T
   * @f}
   * where $C$ is the matrix of coefficients $c_{jk}$ and
   * $X_{ik} = \Psi_i[\tilde\varphi_k]$. Consequently, in order to compute
   * the expansion coefficients $C=X^{-T}$, we need to apply the node
   * functionals to all functions of the "raw" basis of the polynomial space.
   *
   * Until the finite element receives this matrix $X$ back, it describes its
   * shape functions (e.g., in FiniteElement::shape_value()) in the form
   * $\tilde\varphi_j$. After it calls this function, it has the expansion
   * coefficients and can describe its shape functions as $\varphi_j$.
   *
   * This function therefore computes this matrix $X$, for the following
   * specific circumstances:
   * - That the node functionals $\Psi_i$ are point evaluations at points
   *   $\mathbf x_i$ that the finite element in question describes via its
   *   "generalized" support points (through
   *   FiniteElement::get_generalized_support_points(), see also
   *   @ref GlossGeneralizedSupport "this glossary entry"). These point
   *   evaluations need to necessarily evaluate the <i>value</i> of a shape
   *   function at that point (the shape function may be vector-valued, and
   *   so the functional may be a linear combination of the individual
   *   components of the values); but, in particular, the nodal functions may
   *   not be <i>integrals</i> over entire edges or faces,
   *   or other non-local functionals. In other words, we assume that
   *   $\Psi_i[\tilde\varphi_j] = f_j(\tilde\varphi_j(\mathbf x_i))$
   *   where $f_j$ is a function of the (possibly vector-valued) argument
   *   that returns a scalar.
   * - That the finite element has exactly @p dim vector components.
   * - That the function $f_j$ is given by whatever the element implements
   *   through the
   * FiniteElement::convert_generalized_support_point_values_to_dof_values()
   *   function.
   *
   * @param fe The finite element for which the operations above are to be
   *        performed.
   * @return The matrix $X$ as discussed above.
   */
  template <int dim, int spacedim>
  FullMatrix<double>
  compute_node_matrix(const FiniteElement<dim, spacedim> &fe);

  /**
   * For all possible (isotropic and anisotropic) refinement cases compute the
   * embedding matrices from a coarse cell to the child cells. Each column of
   * the resulting matrices contains the representation of a coarse grid basis
   * function by the fine grid basis; the matrices are split such that there
   * is one matrix for every child.
   *
   * This function computes the coarse grid function in a sufficiently large
   * number of quadrature points and fits the fine grid functions using least
   * squares approximation. Therefore, the use of this function is restricted
   * to the case that the finite element spaces are actually nested.
   *
   * Note, that <code>matrices[refinement_case-1][child]</code> includes the
   * embedding (or prolongation) matrix of child <code>child</code> for the
   * RefinementCase <code>refinement_case</code>. Here, we use
   * <code>refinement_case-1</code> instead of <code>refinement_case</code> as
   * for RefinementCase::no_refinement(=0) there are no prolongation matrices
   * available.
   *
   * Typically this function is called by the various implementations of
   * FiniteElement classes in order to fill the respective
   * FiniteElement::prolongation matrices.
   *
   * @param fe The finite element class for which we compute the embedding
   * matrices.
   *
   * @param matrices A reference to RefinementCase<dim>::isotropic_refinement
   * vectors of FullMatrix objects. Each vector corresponds to one
   * RefinementCase @p refinement_case and is of the vector size
   * GeometryInfo<dim>::n_children(refinement_case). This is the format used
   * in FiniteElement, where we want to use this function mostly.
   *
   * @param isotropic_only Set to <code>true</code> if you only want to
   * compute matrices for isotropic refinement.
   *
   * @param threshold is the gap allowed in the least squares algorithm
   * computing the embedding.
   */
  template <int dim, typename number, int spacedim>
  void
  compute_embedding_matrices(
    const FiniteElement<dim, spacedim> &          fe,
    std::vector<std::vector<FullMatrix<number>>> &matrices,
    const bool                                    isotropic_only = false,
    const double                                  threshold      = 1.e-12);

  /**
   * Compute the embedding matrices on faces needed for constraint matrices.
   *
   * @param fe The finite element for which to compute these matrices.
   *
   * @param matrices An array of <i>GeometryInfo<dim>::subfaces_per_face =
   * 2<sup>dim-1</sup></i> FullMatrix objects,holding the embedding matrix for
   * each subface.
   *
   * @param face_coarse The number of the face on the coarse side of the face
   * for which this is computed.
   *
   * @param face_fine The number of the face on the refined side of the face
   * for which this is computed.
   *
   * @param threshold is the gap allowed in the least squares algorithm
   * computing the embedding.
   *
   * @warning This function will be used in computing constraint matrices. It
   * is not sufficiently tested yet.
   */
  template <int dim, typename number, int spacedim>
  void
  compute_face_embedding_matrices(
    const FiniteElement<dim, spacedim> &fe,
    FullMatrix<number> (&matrices)[GeometryInfo<dim>::max_children_per_face],
    const unsigned int face_coarse,
    const unsigned int face_fine,
    const double       threshold = 1.e-12);

  /**
   * For all possible (isotropic and anisotropic) refinement cases compute the
   * <i>L<sup>2</sup></i>-projection matrices from the children to a coarse
   * cell.
   *
   * Note, that <code>matrices[refinement_case-1][child]</code> includes the
   * projection (or restriction) matrix of child <code>child</code> for the
   * RefinementCase <code>refinement_case</code>. Here, we use
   * <code>refinement_case-1</code> instead of <code>refinement_case</code> as
   * for RefinementCase::no_refinement(=0) there are no projection matrices
   * available.
   *
   * Typically this function is called by the various implementations of
   * FiniteElement classes in order to fill the respective
   * FiniteElement::restriction matrices.
   *
   * @arg[in] fe The finite element class for which we compute the projection
   *   matrices.
   *
   * @arg[out] matrices A reference to a set of
   *   <tt>RefinementCase<dim>::isotropic_refinement</tt> vectors of FullMatrix
   *   objects. Each vector corresponds to one RefinementCase @p refinement_case
   *   and is of the vector size
   *   <tt>GeometryInfo<dim>::n_children(refinement_case)</tt>. This is the
   *   format used in FiniteElement, where we want to use this function mostly.
   *
   * @arg[in] isotropic_only If set to <code>true</code>, then this
   *   function only computes data for the isotropic refinement case. The
   *   other elements of the output vector are left untouched (but still
   *   exist).
   */
  template <int dim, typename number, int spacedim>
  void
  compute_projection_matrices(
    const FiniteElement<dim, spacedim> &          fe,
    std::vector<std::vector<FullMatrix<number>>> &matrices,
    const bool                                    isotropic_only = false);

  /**
   * Project scalar data defined in quadrature points to a finite element
   * space on a single cell.
   *
   * What this function does is the following: assume that there is scalar
   * data <tt>u<sub>q</sub>, 0 <= q < Q:=quadrature.size()</tt> defined at the
   * quadrature points of a cell, with the points defined by the given
   * <tt>rhs_quadrature</tt> object. We may then want to ask for that finite
   * element function (on a single cell) <tt>v<sub>h</sub></tt> in the finite-
   * dimensional space defined by the given FE object that is the projection
   * of <tt>u</tt> in the following sense:
   *
   * Usually, the projection <tt>v<sub>h</sub></tt> is that function that
   * satisfies <tt>(v<sub>h</sub>,w)=(u,w)</tt> for all discrete test
   * functions <tt>w</tt>. In the present case, we can't evaluate the right
   * hand side, since <tt>u</tt> is only defined in the quadrature points
   * given by <tt>rhs_quadrature</tt>, so we replace it by a quadrature
   * approximation. Likewise, the left hand side is approximated using the
   * <tt>lhs_quadrature</tt> object; if this quadrature object is chosen
   * appropriately, then the integration of the left hand side can be done
   * exactly, without any approximation. The use of different quadrature
   * objects is necessary if the quadrature object for the right hand side has
   * too few quadrature points -- for example, if data <tt>q</tt> is only
   * defined at the cell center, then the corresponding one-point quadrature
   * formula is obviously insufficient to approximate the scalar product on
   * the left hand side by a definite form.
   *
   * After these quadrature approximations, we end up with a nodal
   * representation <tt>V<sub>h</sub></tt> of <tt>v<sub>h</sub></tt> that
   * satisfies the following system of linear equations: <tt>M V<sub>h</sub> =
   * Q U</tt>, where <tt>M<sub>ij</sub>=(phi_i,phi_j)</tt> is the @ref GlossMassMatrix "mass matrix"
   * approximated by <tt>lhs_quadrature</tt>, and <tt>Q</tt> is the matrix
   * <tt>Q<sub>iq</sub>=phi<sub>i</sub>(x<sub>q</sub>) w<sub>q</sub></tt>
   * where <tt>w<sub>q</sub></tt> are quadrature weights; <tt>U</tt> is the
   * vector of quadrature point data <tt>u<sub>q</sub></tt>.
   *
   * In order to then get the nodal representation <tt>V<sub>h</sub></tt> of
   * the projection of <tt>U</tt>, one computes <tt>V<sub>h</sub> = X U,
   * X=M<sup>-1</sup> Q</tt>. The purpose of this function is to compute the
   * matrix <tt>X</tt> and return it through the last argument of this
   * function.
   *
   * Note that this function presently only supports scalar data. An extension
   * of the mass matrix is of course trivial, but one has to define the order
   * of data in the vector <tt>U</tt> if it contains vector valued data in all
   * quadrature points.
   *
   * A use for this function is described in the introduction to the step-18
   * example program.
   *
   * The opposite of this function, interpolation of a finite element function
   * onto quadrature points is essentially what the
   * <tt>FEValues::get_function_values</tt> functions do; to make things a
   * little simpler, the
   * <tt>FETools::compute_interpolation_to_quadrature_points_matrix</tt>
   * provides the matrix form of this.
   *
   * Note that this function works on a single cell, rather than an entire
   * triangulation. In effect, it therefore doesn't matter if you use a
   * continuous or discontinuous version of the finite element.
   *
   * It is worth noting that there are a few confusing cases of this function.
   * The first one is that it really only makes sense to project onto a finite
   * element that has at most as many degrees of freedom per cell as there are
   * quadrature points; the projection of N quadrature point data into a space
   * with M>N unknowns is well-defined, but often yields funny and non-
   * intuitive results. Secondly, one would think that if the quadrature point
   * data is defined in the support points of the finite element, i.e. the
   * quadrature points of <tt>rhs_quadrature</tt> equal
   * <tt>fe.get_unit_support_points()</tt>, then the projection should be the
   * identity, i.e. each degree of freedom of the finite element equals the
   * value of the given data in the support point of the corresponding shape
   * function. However, this is not generally the case: while the matrix
   * <tt>Q</tt> in that case is the identity matrix, the mass matrix
   * <tt>M</tt> is not equal to the identity matrix, except for the special
   * case that the quadrature formula <tt>lhs_quadrature</tt> also has its
   * quadrature points in the support points of the finite element.
   *
   * Finally, this function only defines a cell wise projection, while one
   * frequently wants to apply it to all cells in a triangulation. However, if
   * it is applied to one cell after the other, the results from later cells
   * may overwrite nodal values computed already from previous cells if
   * degrees of freedom live on the interfaces between cells. The function is
   * therefore most useful for discontinuous elements.
   */
  template <int dim, int spacedim>
  void
  compute_projection_from_quadrature_points_matrix(
    const FiniteElement<dim, spacedim> &fe,
    const Quadrature<dim> &             lhs_quadrature,
    const Quadrature<dim> &             rhs_quadrature,
    FullMatrix<double> &                X);

  /**
   * Given a (scalar) local finite element function, compute the matrix that
   * maps the vector of nodal values onto the vector of values of this
   * function at quadrature points as given by the second argument. In a
   * sense, this function does the opposite of the
   * FETools::compute_projection_from_quadrature_points_matrix function.
   */
  template <int dim, int spacedim>
  void
  compute_interpolation_to_quadrature_points_matrix(
    const FiniteElement<dim, spacedim> &fe,
    const Quadrature<dim> &             quadrature,
    FullMatrix<double> &                I_q);

  /**
   * Compute the projection of tensorial (first-order tensor) data stored at
   * the quadrature points @p vector_of_tensors_at_qp to data @p
   * vector_of_tensors_at_nodes at the support points of the cell.  The data
   * in @p vector_of_tensors_at_qp is ordered sequentially following the
   * quadrature point numbering.  The size of @p vector_of_tensors_at_qp must
   * correspond to the number of columns of @p projection_matrix.  The size of
   * @p vector_of_tensors_at_nodes must correspond to the number of rows of @p
   * vector_of_tensors_at_nodes .  The projection matrix @p projection_matrix
   * describes the projection of scalar data from the quadrature points and
   * can be obtained from the
   * FETools::compute_projection_from_quadrature_points_matrix function.
   */
  template <int dim>
  void
  compute_projection_from_quadrature_points(
    const FullMatrix<double> &         projection_matrix,
    const std::vector<Tensor<1, dim>> &vector_of_tensors_at_qp,
    std::vector<Tensor<1, dim>> &      vector_of_tensors_at_nodes);



  /**
   * same as last function but for a @p SymmetricTensor .
   */
  template <int dim>
  void
  compute_projection_from_quadrature_points(
    const FullMatrix<double> &                  projection_matrix,
    const std::vector<SymmetricTensor<2, dim>> &vector_of_tensors_at_qp,
    std::vector<SymmetricTensor<2, dim>> &      vector_of_tensors_at_nodes);



  /**
   * This method implements the
   * FETools::compute_projection_from_quadrature_points_matrix method for
   * faces of a mesh.  The matrix that it returns, X, is face specific and its
   * size is fe.n_dofs_per_cell() by rhs_quadrature.size().  The dimension, dim
   * must be larger than 1 for this class, since Quadrature<dim-1> objects are
   * required. See the documentation on the Quadrature class for more
   * information.
   */
  template <int dim, int spacedim>
  void
  compute_projection_from_face_quadrature_points_matrix(
    const FiniteElement<dim, spacedim> &fe,
    const Quadrature<dim - 1> &         lhs_quadrature,
    const Quadrature<dim - 1> &         rhs_quadrature,
    const typename DoFHandler<dim, spacedim>::active_cell_iterator &cell,
    const unsigned int                                              face,
    FullMatrix<double> &                                            X);



  /**
   * Wrapper around
   * FiniteElement::convert_generalized_support_point_values_to_dof_values()
   * that works with arbitrary number types.
   *
   * @param[in] finite_element The FiniteElement to compute dof values for.
   * @param[in] support_point_values An array of size @p dofs_per_cell
   *   (which equals the number of points the get_generalized_support_points()
   *   function will return) where each element is a vector with as many entries
   *   as the element has vector components. This array should contain
   *   the values of a function at the generalized support points of the
   *   finite element.
   * @param[out] dof_values An array of size @p dofs_per_cell that contains
   *   the node functionals of the element applied to the given function.
   */
  template <int dim, int spacedim, typename number>
  void
  convert_generalized_support_point_values_to_dof_values(
    const FiniteElement<dim, spacedim> &finite_element,
    const std::vector<Vector<number>> & support_point_values,
    std::vector<number> &               dof_values);



  /** @} */
  /**
   * @name Functions which should be in DoFTools
   */
  /** @{ */
  /**
   * Compute the interpolation of a the @p dof1-function @p u1 to a @p
   * dof2-function @p u2. @p dof1 and @p dof2 need to be DoFHandlers based on
   * the same triangulation.
   *
   * If the elements @p fe1 and @p fe2 are either both continuous or both
   * discontinuous then this interpolation is the usual point interpolation.
   * The same is true if @p fe1 is a continuous and @p fe2 is a discontinuous
   * finite element. For the case that @p fe1 is a discontinuous and @p fe2 is
   * a continuous finite element there is no point interpolation defined at
   * the discontinuities.  Therefore the mean value is taken at the DoF values
   * on the discontinuities.
   *
   * Note that for continuous elements on grids with hanging nodes (i.e.
   * locally refined grids) this function does not give the expected output.
   * Indeed, the resulting output vector does not necessarily respect
   * continuity requirements at hanging nodes: if, for example, you are
   * interpolating a Q2 field to a Q1 field, then at hanging nodes the output
   * field will have the function value of the input field, which however is
   * not usually the mean value of the two adjacent nodes. It is thus not part
   * of the Q1 function space on the whole triangulation, although it is of
   * course Q1 on each cell.
   *
   * For this case (continuous elements on grids with hanging nodes), please
   * use the @p interpolate() function with an additional AffineConstraints
   * object as argument, see below, or make the field conforming yourself
   * by calling the @p distribute function of your hanging node constraints
   * object.
   */
  template <int dim, int spacedim, class InVector, class OutVector>
  void
  interpolate(const DoFHandler<dim, spacedim> &dof1,
              const InVector &                 u1,
              const DoFHandler<dim, spacedim> &dof2,
              OutVector &                      u2);

  /**
   * Compute the interpolation of a the @p dof1-function @p u1 to a @p
   * dof2-function @p u2. @p dof1 and @p dof2 need to be DoFHandlers based on
   * the same triangulation. @p constraints is a hanging node constraints object
   * corresponding to @p dof2. This object is particular important when
   * interpolating onto continuous elements on grids with hanging nodes (locally
   * refined grids).
   *
   * If the elements @p fe1 and @p fe2 are either both continuous or both
   * discontinuous then this interpolation is the usual point interpolation.
   * The same is true if @p fe1 is a continuous and @p fe2 is a discontinuous
   * finite element. For the case that @p fe1 is a discontinuous and @p fe2 is
   * a continuous finite element there is no point interpolation defined at
   * the discontinuities.  Therefore the mean value is taken at the DoF values
   * at the discontinuities.
   */
  template <int dim, int spacedim, class InVector, class OutVector>
  void
  interpolate(
    const DoFHandler<dim, spacedim> &                        dof1,
    const InVector &                                         u1,
    const DoFHandler<dim, spacedim> &                        dof2,
    const AffineConstraints<typename OutVector::value_type> &constraints,
    OutVector &                                              u2);

  /**
   * Compute the interpolation of the @p fe1-function @p u1 to a @p
   * fe2-function, and interpolates this to a second @p fe1-function named @p
   * u1_interpolated.
   *
   * Note, that this function does not work on continuous elements at hanging
   * nodes. For that case use the @p back_interpolate function, below, that
   * takes an additional @p AffineConstraints object.
   *
   * Furthermore note, that for the specific case when the finite element
   * space corresponding to @p fe1 is a subset of the finite element space
   * corresponding to @p fe2, this function is simply an identity mapping.
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  back_interpolate(const DoFHandler<dim, spacedim> &   dof1,
                   const InVector &                    u1,
                   const FiniteElement<dim, spacedim> &fe2,
                   OutVector &                         u1_interpolated);

  /**
   * Compute the interpolation of the @p dof1-function @p u1 to a @p
   * dof2-function, and interpolates this to a second @p dof1-function named
   * @p u1_interpolated.  @p constraints1 and @p constraints2 are the hanging
   * node constraints corresponding to @p dof1 and @p dof2, respectively.
   * These objects are particular important when continuous elements on grids
   * with hanging nodes (locally refined grids) are involved.
   *
   * Furthermore note, that for the specific case when the finite element
   * space corresponding to @p dof1 is a subset of the finite element space
   * corresponding to @p dof2, this function is simply an identity mapping.
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  back_interpolate(
    const DoFHandler<dim, spacedim> &                        dof1,
    const AffineConstraints<typename OutVector::value_type> &constraints1,
    const InVector &                                         u1,
    const DoFHandler<dim, spacedim> &                        dof2,
    const AffineConstraints<typename OutVector::value_type> &constraints2,
    OutVector &                                              u1_interpolated);

  /**
   * Compute $(Id-I_h)z_1$ for a given @p dof1-function $z_1$, where $I_h$ is
   * the interpolation from @p fe1 to @p fe2. The result $(Id-I_h)z_1$ is
   * written into @p z1_difference.
   *
   * Note, that this function does not work for continuous elements at hanging
   * nodes. For that case use the @p interpolation_difference function, below,
   * that takes an additional @p AffineConstraints object.
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  interpolation_difference(const DoFHandler<dim, spacedim> &   dof1,
                           const InVector &                    z1,
                           const FiniteElement<dim, spacedim> &fe2,
                           OutVector &                         z1_difference);

  /**
   * Compute $(Id-I_h)z_1$ for a given @p dof1-function $z_1$, where $I_h$ is
   * the interpolation from @p fe1 to @p fe2. The result $(Id-I_h)z_1$ is
   * written into @p z1_difference.  @p constraints1 and @p constraints2 are
   * the hanging node constraints corresponding to @p dof1 and @p dof2,
   * respectively. These objects are particular important when continuous
   * elements on grids with hanging nodes (locally refined grids) are
   * involved.
   *
   * For parallel computations, supply @p z1 with ghost elements and @p
   * z1_difference without ghost elements.
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  interpolation_difference(
    const DoFHandler<dim, spacedim> &                        dof1,
    const AffineConstraints<typename OutVector::value_type> &constraints1,
    const InVector &                                         z1,
    const DoFHandler<dim, spacedim> &                        dof2,
    const AffineConstraints<typename OutVector::value_type> &constraints2,
    OutVector &                                              z1_difference);



  /**
   * $L^2$ projection for discontinuous elements. Operates the same direction
   * as interpolate.
   *
   * The global projection can be computed by local matrices if the finite
   * element spaces are discontinuous. With continuous elements, this is
   * impossible, since a global @ref GlossMassMatrix "mass matrix" must be inverted.
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  project_dg(const DoFHandler<dim, spacedim> &dof1,
             const InVector &                 u1,
             const DoFHandler<dim, spacedim> &dof2,
             OutVector &                      u2);

  /**
   * Compute the patchwise extrapolation of a @p dof1 function @p z1 to a @p
   * dof2 function @p z2.  @p dof1 and @p dof2 need to be DoFHandler objects
   * based on the same triangulation. This function is used, for example, for
   * extrapolating patchwise a piecewise linear solution to a piecewise
   * quadratic solution.
   *
   * The function's name is historical and probably not particularly well
   * chosen. The function performs the following operations, one after the
   * other:
   *
   * - It interpolates directly from every cell of @p dof1 to the
   * corresponding cell of `dof2` using the interpolation matrix of the finite
   * element spaces used on these cells and provided by the finite element
   * objects involved. This step is done using the FETools::interpolate()
   * function.
   * - It then performs a loop over all non-active cells of `dof2`.
   * If such a non-active cell has at least one active child, then we call the
   * children of this cell a "patch". We then interpolate from the children of
   * this patch to the patch, using the finite element space associated with
   * `dof2` and immediately interpolate back to the children. In essence, this
   * information throws away all information in the solution vector that lives
   * on a scale smaller than the patch cell.
   * - Since we traverse non-active cells from the coarsest to the finest
   * levels, we may find patches that correspond to child cells of previously
   * treated patches if the mesh had been refined adaptively (this cannot
   * happen if the  mesh has been refined globally because there the children
   * of a patch are all active). We also perform the operation described above
   * on these patches, but it is easy to see that on patches that are children
   * of previously treated patches, the operation is now the identity operation
   * (since it interpolates from the children of the current patch a function
   * that had previously been interpolated to these children from an even
   * coarser patch). Consequently, this does not alter the solution vector any
   * more.
   *
   * The name of the function originates from the fact that it can be used to
   * construct a representation of a function of higher polynomial degree on a
   * once coarser mesh. For example, if you imagine that you start with a
   * $Q_1$ function on a globally refined mesh, and that @p dof2 is associated
   * with a $Q_2$ element, then this function computes the equivalent of the
   * operator $I_{2h}^{(2)}$ interpolating the original piecewise linear
   * function onto a quadratic function on a once coarser mesh with mesh size
   * $2h$ (but representing this function on the original mesh with size $h$).
   * If the exact solution is sufficiently smooth, then
   * $u^\ast=I_{2h}^{(2)}u_h$ is typically a better approximation to the exact
   * solution $u$ of the PDE than $u_h$ is. In other words, this function
   * provides a postprocessing step that improves the solution in a similar
   * way one often obtains by extrapolating a sequence of solutions,
   * explaining the origin of the function's name.
   *
   * @note The resulting field does not satisfy continuity requirements of the
   * given finite elements if the algorithm outlined above is used. When you
   * use continuous elements on grids with hanging nodes, please use the @p
   * extrapolate function with an additional AffineConstraints argument, see
   * below.
   *
   * @note Since this function operates on patches of cells, it requires that
   * the underlying grid is refined at least once for every coarse grid cell.
   * If this is not the case, an exception will be raised.
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  extrapolate(const DoFHandler<dim, spacedim> &dof1,
              const InVector &                 z1,
              const DoFHandler<dim, spacedim> &dof2,
              OutVector &                      z2);

  /**
   * Compute the patchwise extrapolation of a @p dof1 function @p z1 to a @p
   * dof2 function @p z2.  @p dof1 and @p dof2 need to be DoFHandler objects
   * based on the same triangulation.  @p constraints is a hanging node
   * constraints object corresponding to @p dof2. This object is necessary
   * when interpolating onto continuous elements on grids with hanging nodes
   * (locally refined grids).
   *
   * Otherwise, the function does the same as the other @p extrapolate
   * function above (for which the documentation provides an extensive
   * description of its operation).
   */
  template <int dim, class InVector, class OutVector, int spacedim>
  void
  extrapolate(
    const DoFHandler<dim, spacedim> &                        dof1,
    const InVector &                                         z1,
    const DoFHandler<dim, spacedim> &                        dof2,
    const AffineConstraints<typename OutVector::value_type> &constraints,
    OutVector &                                              z2);

  /** @} */
  /**
   * The numbering of the degrees of freedom in continuous finite elements is
   * hierarchic, i.e. in such a way that we first number the vertex dofs, in
   * the order of the vertices as defined by the triangulation, then the line
   * dofs in the order and respecting the direction of the lines, then the
   * dofs on quads, etc. However, we could have, as well, numbered them in a
   * lexicographic way, i.e. with indices first running in x-direction, then
   * in y-direction and finally in z-direction. Discontinuous elements of
   * class FE_DGQ() are numbered in this way, for example.
   *
   * This function returns a vector containing information about the
   * lexicographic index each degree of freedom in the hierarchic numbering
   * would have to a given degree of a continuous finite element.
   */
  template <int dim>
  std::vector<unsigned int>
  hierarchic_to_lexicographic_numbering(unsigned int degree);

  /**
   * This is the reverse function to the above one, generating the map from
   * the lexicographic to the hierarchical numbering for a given polynomial
   * degree of a continuous finite element. All the remarks made about the
   * above function are also valid here.
   */
  template <int dim>
  std::vector<unsigned int>
  lexicographic_to_hierarchic_numbering(unsigned int degree);

  /**
   * A namespace that contains functions that help setting up internal
   * data structures when implementing FiniteElement which are build
   * from simpler ("base") elements, for example FESystem. The things
   * computed by these functions typically serve as constructor
   * arguments to the FiniteElement base class of the derived finite
   * element object being constructed.
   *
   * There are generally two ways in which one can build more complex
   * elements, and this is reflected by several of the functions in
   * this namespace having arguments called
   * <code>do_tensor_product</code>:
   *
   * <ol>
   * <li> Tensor product construction (<code>do_tensor_product=true</code>):
   * The tensor product construction, in the simplest case, builds a
   * vector-valued element from scalar elements (see
   * @ref vector_valued "this documentation module"
   * and
   * @ref GlossComponent "this glossary entry"
   * for more information).
   * To give an example, consider creating a vector-valued element with
   * two vector components, where the first should have linear shape
   * functions and the second quadratic shape functions. In 1d, the
   * shape functions (on the reference cell) of the base elements are then
   * @f{align*}{
   *   Q_1 &= \{ 1-x, x \},
   *   \\  Q_2 &= \{ 2(\frac 12 - x)(1-x), 2(x - \frac 12)x, 4x(1-x) \},
   * @f}
   * where shape functions are ordered in the usual way (first on the
   * first vertex, then on the second vertex, then in the interior of
   * the cell). The tensor product construction will create an element with
   * the following shape functions:
   * @f{align*}{
   *   Q_1 \times Q_2 &=
   *   \left\{
   *     \begin{pmatrix} 1-x \\ 0 \end{pmatrix},
   *     \begin{pmatrix} 0 \\ 2(\frac 12 - x)(1-x)  \end{pmatrix},
   *     \begin{pmatrix} x \\ 0 \end{pmatrix},
   *     \begin{pmatrix} 0 \\ 2(x - \frac 12)x \end{pmatrix},
   *     \begin{pmatrix} 0 \\ 4x(1-x) \end{pmatrix}
   *   \right\}.
   * @f}
   * The list here is again in standard order.
   *
   * Of course, the procedure also works if the base elements are
   * already vector valued themselves: in that case, the composed
   * element simply has as many vector components as the base elements
   * taken together.
   *
   * <li> Combining shape functions
   * (<code>do_tensor_product=false</code>): In contrast to the
   * previous strategy, combining shape functions simply takes
   * <i>all</i> of the shape functions together. In the case above,
   * this would yield the following element:
   * @f{align*}{
   *   Q_1 + Q_2 &= \{ 1-x, 2(\frac 12 - x)(1-x),
   *                   x, 2(x - \frac 12)x, 4x(1-x) \}.
   * @f}
   * In other words, if the base elements are scalar, the resulting
   * element will also be. In general, the base elements all will
   * have to have the same number of vector components.
   *
   * The element constructed above of course no longer has a linearly
   * independent set of shape functions. As a consequence, any matrix
   * one creates by treating all shape functions of the composed
   * element in the same way will be singular. In practice, this
   * strategy is therefore typically used in situations where one
   * explicitly makes sure that certain shape functions are treated
   * differently (e.g., by multiplying them with weight functions), or
   * in cases where the shape functions one combines are not linearly
   * dependent.
   *
   * </ol>
   */
  namespace Compositing
  {
    /**
     * Take vectors of finite elements and multiplicities and multiply out
     * how many degrees of freedom the composed element has per vertex,
     * line, etc.
     *
     * If @p do_tensor_product is true, the number of components
     * returned in the FiniteElementData object is the sum over the
     * product of the number of components in each of the finite
     * elements times the corresponding multiplicity.  Otherwise the
     * number of components is taken from the first finite element with
     * non-zero multiplicity, and all other elements with non-zero
     * multiplicities need to have the same number of vector components.
     *
     * See the documentation of namespace FETools::Compositing for more
     * information about the @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    FiniteElementData<dim>
    multiply_dof_numbers(
      const std::vector<const FiniteElement<dim, spacedim> *> &fes,
      const std::vector<unsigned int> &                        multiplicities,
      const bool do_tensor_product = true);

    /**
     * Same as above for an arbitrary number of parameters of type
     * <code>std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned
     * int></code> and <code>do_tensor_product = true</code>.
     */
    template <int dim, int spacedim>
    FiniteElementData<dim>
    multiply_dof_numbers(
      const std::initializer_list<
        std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>>
        &fe_systems);

    /**
     * Same as above but for a specific number of sub-elements.
     */
    template <int dim, int spacedim>
    FiniteElementData<dim>
    multiply_dof_numbers(const FiniteElement<dim, spacedim> *fe1,
                         const unsigned int                  N1,
                         const FiniteElement<dim, spacedim> *fe2 = nullptr,
                         const unsigned int                  N2  = 0,
                         const FiniteElement<dim, spacedim> *fe3 = nullptr,
                         const unsigned int                  N3  = 0,
                         const FiniteElement<dim, spacedim> *fe4 = nullptr,
                         const unsigned int                  N4  = 0,
                         const FiniteElement<dim, spacedim> *fe5 = nullptr,
                         const unsigned int                  N5  = 0);

    /**
     * Compute the "restriction is additive" flags (see the
     * documentation of the FiniteElement class) for a list of finite
     * elements with multiplicities given in the second argument.
     *
     * The "restriction is additive" flags are properties of
     * individual shape functions that do not depend on whether the
     * composed element uses the tensor product or combination
     * strategy outlined in the documentation of the
     * FETools::Composition namespace. Consequently, this function
     * does not have a @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    std::vector<bool>
    compute_restriction_is_additive_flags(
      const std::vector<const FiniteElement<dim, spacedim> *> &fes,
      const std::vector<unsigned int> &                        multiplicities);

    /**
     * Same as above for an arbitrary number of parameters of type
     * <code>std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned
     * int></code>.
     */
    template <int dim, int spacedim>
    std::vector<bool>
    compute_restriction_is_additive_flags(
      const std::initializer_list<
        std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>>
        &fe_systems);

    /**
     * Take a @p FiniteElement object and return a boolean vector
     * describing the @p restriction_is_additive_flags (see the
     * documentation of the FiniteElement class) for each shape function
     * of the mixed element consisting of @p N1, @p N2, ... copies of
     * the sub-elements @p fe1, @p fe2, ...
     *
     * The "restriction is additive" flags are properties of
     * individual shape functions that do not depend on whether the
     * composed element uses the tensor product or combination
     * strategy outlined in the documentation of the
     * FETools::Composition namespace. Consequently, this function
     * does not have a @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    std::vector<bool>
    compute_restriction_is_additive_flags(
      const FiniteElement<dim, spacedim> *fe1,
      const unsigned int                  N1,
      const FiniteElement<dim, spacedim> *fe2 = nullptr,
      const unsigned int                  N2  = 0,
      const FiniteElement<dim, spacedim> *fe3 = nullptr,
      const unsigned int                  N3  = 0,
      const FiniteElement<dim, spacedim> *fe4 = nullptr,
      const unsigned int                  N4  = 0,
      const FiniteElement<dim, spacedim> *fe5 = nullptr,
      const unsigned int                  N5  = 0);


    /**
     * Compute the nonzero components for each shape function of a
     * composed finite element described by a list of finite elements
     * with multiplicities given in the second argument.
     *
     * If @p do_tensor_product is true, the number of components (and
     * thus the size of the ComponentMask objects) is the sum over the
     * product of the number of components in each of the finite
     * elements times the corresponding multiplicity.  Otherwise the
     * number of components is taken from the first finite element with
     * non-zero multiplicity, and all other elements with non-zero
     * multiplicities need to have the same number of vector components.
     *
     * See the documentation of namespace FETools::Compositing for more
     * information about the @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    std::vector<ComponentMask>
    compute_nonzero_components(
      const std::vector<const FiniteElement<dim, spacedim> *> &fes,
      const std::vector<unsigned int> &                        multiplicities,
      const bool do_tensor_product = true);

    /**
     * Same as above for an arbitrary number of parameters of type
     * <code>std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned
     * int></code> and <code>do_tensor_product = true</code>.
     */
    template <int dim, int spacedim>
    std::vector<ComponentMask>
    compute_nonzero_components(
      const std::initializer_list<
        std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>>
        &fe_systems);

    /**
     * Compute the non-zero vector components of a composed finite
     * element. This function is similar to the previous one, except
     * that the pointers indicate the elements to be composed, and the
     * arguments @p N1, @p N2, ... the multiplicities. Null pointers
     * indicate that an argument is to be skipped.
     *
     * If @p do_tensor_product is true, the number of components (and
     * thus the size of the ComponentMask objects) is the sum over the
     * product of the number of components in each of the finite
     * elements times the corresponding multiplicity.  Otherwise the
     * number of components is taken from the first finite element with
     * non-zero multiplicity, and all other elements with non-zero
     * multiplicities need to have the same number of vector components.
     *
     * See the documentation of namespace FETools::Compositing for more
     * information about the @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    std::vector<ComponentMask>
    compute_nonzero_components(
      const FiniteElement<dim, spacedim> *fe1,
      const unsigned int                  N1,
      const FiniteElement<dim, spacedim> *fe2               = nullptr,
      const unsigned int                  N2                = 0,
      const FiniteElement<dim, spacedim> *fe3               = nullptr,
      const unsigned int                  N3                = 0,
      const FiniteElement<dim, spacedim> *fe4               = nullptr,
      const unsigned int                  N4                = 0,
      const FiniteElement<dim, spacedim> *fe5               = nullptr,
      const unsigned int                  N5                = 0,
      const bool                          do_tensor_product = true);

    /**
     * For a given (composite) @p finite_element build @p
     * system_to_component_table, @p system_to_base_table and @p
     * component_to_base_table.
     *
     * If @p do_tensor_product is true, the number of components
     * used for the composite element is the sum over the
     * product of the number of components in each of the finite
     * elements times the corresponding multiplicity.  Otherwise the
     * number of components is taken from the first finite element with
     * non-zero multiplicity, and all other elements with non-zero
     * multiplicities need to have the same number of vector components.
     *
     * See the documentation of namespace FETools::Compositing for more
     * information about the @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    void
    build_cell_tables(
      std::vector<std::pair<std::pair<unsigned int, unsigned int>,
                            unsigned int>> &system_to_base_table,
      std::vector<std::pair<unsigned int, unsigned int>>
        &                                   system_to_component_table,
      std::vector<std::pair<std::pair<unsigned int, unsigned int>,
                            unsigned int>> &component_to_base_table,
      const FiniteElement<dim, spacedim> &  finite_element,
      const bool                            do_tensor_product = true);

    /**
     * For a given (composite) @p finite_element build @p face_system_to_base_table,
     * and @p face_system_to_component_table.
     *
     * If @p do_tensor_product is true, the number of components
     * used for the composite element is the sum over the
     * product of the number of components in each of the finite
     * elements times the corresponding multiplicity.  Otherwise the
     * number of components is taken from the first finite element with
     * non-zero multiplicity, and all other elements with non-zero
     * multiplicities need to have the same number of vector components.
     *
     * See the documentation of namespace FETools::Compositing for more
     * information about the @p do_tensor_product argument.
     */
    template <int dim, int spacedim>
    void
    build_face_tables(
      std::vector<std::pair<std::pair<unsigned int, unsigned int>,
                            unsigned int>> &face_system_to_base_table,
      std::vector<std::pair<unsigned int, unsigned int>>
        &                                 face_system_to_component_table,
      const FiniteElement<dim, spacedim> &finite_element,
      const bool                          do_tensor_product = true,
      const unsigned int                  face_no           = 0 /*TODO*/);

  } // namespace Compositing


  /**
   * Parse the name of a finite element and generate a finite element object
   * accordingly. The parser ignores space characters between words (things
   * matching the regular expression [A-Za-z0-9_]).
   *
   * The name must be in the form which is returned by the
   * FiniteElement::get_name function, where dimension template parameters
   * &lt;2&gt; etc. can be omitted. Alternatively, the explicit number can be
   * replaced by <tt>dim</tt> or <tt>d</tt>. If a number is given, it
   * <b>must</b> match the template parameter of this function.
   *
   * The names of FESystem elements follow the pattern
   * <code>FESystem[FE_Base1^p1-FE_Base2^p2]</code> The powers <code>p1</code>
   * etc. may either be numbers or can be replaced by <tt>dim</tt> or
   * <tt>d</tt>.
   *
   *
   * If no finite element can be reconstructed from this string, an exception
   * of type @p FETools::ExcInvalidFEName is thrown.
   *
   * The function returns a std::unique_ptr to a newly created finite element
   * meaning the caller obtains ownership over the returned object.
   *
   * Since the value of the template argument can't be deduced from the
   * (string) argument given to this function, you have to explicitly specify
   * it when you call this function.
   *
   * This function knows about all the standard elements defined in the
   * library. However, it doesn't by default know about elements that you may
   * have defined in your program. To make your own elements known to this
   * function, use the add_fe_name() function.  This function does not work if
   * one wants to get a codimension 1 finite element.
   */
  template <int dim, int spacedim = dim>
  std::unique_ptr<FiniteElement<dim, spacedim>>
  get_fe_by_name(const std::string &name);

  /**
   * Extend the list of finite elements that can be generated by
   * get_fe_by_name() by the one given as @p name. If get_fe_by_name() is
   * later called with this name, it will use the object given as second
   * argument to create a finite element object.
   *
   * The format of the @p name parameter should include the name of a finite
   * element. However, it is safe to use either the class name alone or to use
   * the result of FiniteElement::get_name (which includes the space dimension
   * as well as the polynomial degree), since everything after the first non-
   * name character will be ignored.
   *
   * The FEFactory object should be an object newly created with <tt>new</tt>.
   * FETools will take ownership of this object and delete it once it is not
   * used anymore.
   *
   * In most cases, if you want objects of type <code>MyFE</code> be created
   * whenever the name <code>my_fe</code> is given to get_fe_by_name, you
   * will want the second argument to this function be of type
   * FEFactory@<MyFE@>, but you can of course create your custom finite
   * element factory class.
   *
   * This function takes over ownership of the object given as second
   * argument, i.e. you should never attempt to destroy it later on. The
   * object will be deleted at the end of the program's lifetime.
   *
   * If the name of the element is already in use, an exception is thrown.
   * Thus, functionality of get_fe_by_name() can only be added, not changed.
   *
   * @note This function manipulates a global table (one table for each space
   * dimension). It is thread safe in the sense that every access to this
   * table is secured by a lock. Nevertheless, since each name can be added
   * only once, user code has to make sure that only one thread adds a new
   * element.
   *
   * Note also that this table exists once for each space dimension. If you
   * have a program that works with finite elements in different space
   * dimensions (for example,
   * @ref step_4 "step-4"
   * does something like this), then you should call this function for each
   * space dimension for which you want your finite element added to the map.
   */
  template <int dim, int spacedim>
  void
  add_fe_name(const std::string &                 name,
              const FEFactoryBase<dim, spacedim> *factory);

  /**
   * The string used for get_fe_by_name() cannot be translated to a finite
   * element.
   *
   * Either the string is badly formatted or you are using a custom element
   * that must be added using add_fe_name() first.
   *
   * @ingroup Exceptions
   */
  DeclException1(ExcInvalidFEName,
                 std::string,
                 << "Can't re-generate a finite element from the string '"
                 << arg1 << "'.");

  /**
   * The string used for get_fe_by_name() cannot be translated to a finite
   * element.
   *
   * Dimension arguments in finite element names should be avoided. If they
   * are there, the dimension should be <tt>dim</tt> or <tt>d</tt>. Here, you
   * gave a numeric dimension argument, which does not match the template
   * dimension of the finite element class.
   *
   * @ingroup Exceptions
   */
  DeclException2(ExcInvalidFEDimension,
                 char,
                 int,
                 << "The dimension " << arg1
                 << " in the finite element string must match "
                 << "the space dimension " << arg2 << '.');

  /**
   * Exception
   *
   * @ingroup Exceptions
   */
  DeclException0(ExcInvalidFE);

  /**
   * The finite element must be
   * @ref GlossPrimitive "primitive".
   *
   * @ingroup Exceptions
   */
  DeclException0(ExcFENotPrimitive);
  /**
   * Exception
   *
   * @ingroup Exceptions
   */
  DeclException0(ExcTriangulationMismatch);

  /**
   * A continuous element is used on a mesh with hanging nodes, but the
   * constraint matrices are missing.
   *
   * @ingroup Exceptions
   */
  DeclExceptionMsg(ExcHangingNodesNotAllowed,
                   "You are using continuous elements on a grid with "
                   "hanging nodes but without providing hanging node "
                   "constraints. Use the respective function with "
                   "additional AffineConstraints argument(s), instead.");
  /**
   * You need at least two grid levels.
   *
   * @ingroup Exceptions
   */
  DeclException0(ExcGridNotRefinedAtLeastOnce);
  /**
   * The dimensions of the matrix used did not match the expected dimensions.
   *
   * @ingroup Exceptions
   */
  DeclException4(ExcMatrixDimensionMismatch,
                 int,
                 int,
                 int,
                 int,
                 << "This is a " << arg1 << 'x' << arg2 << " matrix, "
                 << "but should be a " << arg3 << 'x' << arg4 << " matrix.");

  /**
   * Exception thrown if an embedding matrix was computed inaccurately.
   *
   * @ingroup Exceptions
   */
  DeclException1(ExcLeastSquaresError,
                 double,
                 << "Least squares fit leaves a gap of " << arg1);

  /**
   * Exception thrown if one variable may not be greater than another.
   *
   * @ingroup Exceptions
   */
  DeclException2(ExcNotGreaterThan,
                 int,
                 int,
                 << arg1 << " must be greater than " << arg2);
} // namespace FETools


#ifndef DOXYGEN

namespace FETools
{
  template <class FE>
  std::unique_ptr<FiniteElement<FE::dimension, FE::space_dimension>>
  FEFactory<FE>::get(const unsigned int degree) const
  {
    return std::make_unique<FE>(degree);
  }

  namespace Compositing
  {
    template <int dim, int spacedim>
    std::vector<bool>
    compute_restriction_is_additive_flags(
      const std::initializer_list<
        std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>>
        &fe_systems)
    {
      std::vector<const FiniteElement<dim, spacedim> *> fes;
      std::vector<unsigned int>                         multiplicities;

      const auto extract =
        [&fes, &multiplicities](
          const std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>,
                          unsigned int> &fe_system) {
          fes.push_back(fe_system.first.get());
          multiplicities.push_back(fe_system.second);
        };

      for (const auto &p : fe_systems)
        extract(p);

      return compute_restriction_is_additive_flags(fes, multiplicities);
    }



    template <int dim, int spacedim>
    FiniteElementData<dim>
    multiply_dof_numbers(
      const std::initializer_list<
        std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>>
        &fe_systems)
    {
      std::vector<const FiniteElement<dim, spacedim> *> fes;
      std::vector<unsigned int>                         multiplicities;

      const auto extract =
        [&fes, &multiplicities](
          const std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>,
                          unsigned int> &fe_system) {
          fes.push_back(fe_system.first.get());
          multiplicities.push_back(fe_system.second);
        };

      for (const auto &p : fe_systems)
        extract(p);

      return multiply_dof_numbers(fes, multiplicities, true);
    }



    template <int dim, int spacedim>
    std::vector<ComponentMask>
    compute_nonzero_components(
      const std::initializer_list<
        std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>>
        &fe_systems)
    {
      std::vector<const FiniteElement<dim, spacedim> *> fes;
      std::vector<unsigned int>                         multiplicities;

      const auto extract =
        [&fes, &multiplicities](
          const std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>,
                          unsigned int> &fe_system) {
          fes.push_back(fe_system.first.get());
          multiplicities.push_back(fe_system.second);
        };

      for (const auto &p : fe_systems)
        extract(p);

      return compute_nonzero_components(fes, multiplicities, true);
    }
  } // namespace Compositing
} // namespace FETools

#endif

/** @} */

DEAL_II_NAMESPACE_CLOSE

#endif /* dealii_fe_tools_H */
